I am having a little bit of trouble with the following:
$$\int_{\gamma}\frac{z^2-1}{z^2+1}dz$$ where $\gamma$ is a circle of radius $2$ centered at 0. I am trying to separate this or simplify it into ...

I'm looking for various ways to evaluate the integral:
$$\int_0^\infty \sin x\sin \sqrt{x}\,dx$$
I'm mainly interested in complex analysis. I can think of a wedge -shaped contour of angle $\pi/4$ but ...

Use Cauchy's integral formula to compute the following:
$$\int \limits_{\Gamma} \frac{\cos(z)+i\sin(z)}{(z^2+36)(z+2)}dz$$ where $\Gamma$ is the circle of centre $0$ and radius $3$ traversed in the ...

How do I integrate $\int_{0}^{\infty}\frac{\cos(ax)-\cos(bx)}{x^2}\text{d}x$, for positive and real $a,b$?
I know the contour that I have to use is a semicircle with a small semicircle cut out near ...

What would the criteria on the variable $v$ be such that $f\left( t\right) $
is always negative .
$$f\left( t\right) =\int_{\mathbb{R}^{+}}\frac{\cos \left( xt\right) }{x^{v}}%
dx=\frac{\Gamma \left( ...

I am looking for a particular form of an integral which some simplified version of it has the following form
$$
\Im\int_{0}^{\infty} \frac{\sqrt{1+u^4-u^6}}{u^5}du.
$$
Could someone gives some idea ...

The complex form of the equation for an ellipse with foci at 1 and -1 is $|z-1|+|z+1|=\sqrt{8}$.
a) Find the values of $a$ and $b$ such that $x^2/a^2+y^2/b^2=1$ describe the same ellipse.
b) Let $C$ ...

Let $C$ be the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$. What is the length of $C$?
Would the length of $C$ be $5$ or $10$? I think $r=5$ so I am not sure whether that would be the ...

From Rudin's book, we are to calculate $\int_\mathbb{R} \Big(\frac{\sin x}{x}\Big)^2 e^{itx}dx$ where $i$ is the imaginary number and $t\in\mathbb{R}$.
I'm looking for a hint on how to get started. I ...